Question

A bag contains 7 blue marbles, 7 yellow marbles, and 8 red marbles. Ameer reaches in and pulls out a marble, without Ameer replacing their marble, Becky reaches in and grabs a marble. What is the probability that they both pull out a yellow marble? Enter your answer as a decimal rounded to 3 decimal places. 2. A bag contains 18 blue marbles, 10 green marbles, and 11 yellow marbles. Archie reaches in and grabs out a marble, looks at it, then puts it back in the bag, then Betty does the same thing. What is the probability that a blue and green marble are pulled out (in either order)? Round your answer to 3 decimal places. 3. A quiz consists of 7 multiple choice questions, each with three possible answers, and 4 true-false questions. How many ways can all of the questions on the quiz be answered? 4. A university club has 30 members. How many ways can the club choose three members to attend a university training? 5. A bowl contains 10 pieces of plastic fruit, 5 are pears and the rest are apples. If all of the pears are identical and all of the apples are identical, how many ways can you line up all of the fruit in the bowl? 6. The expected value of playing a game called Moola at K casino is −$0.03 for the person playing the game. If last Saturday 7678 games of Moola are played at K casino, what should K Casino expect to gain from just the game of Moola last Saturday? You do not use the $ symbol in your answer. Round your answer to the nearest cent if applicable. 7. The probability of Juan making a free throw is 71/85 in a basketball game. What is the probability that Juan does not make his next free throw? Round your answer to three decimal places. 8. A university club has 19 members. How many ways can the club choose a President, VicePresident, and Treasurer? Two fair (not weighted) six sided dice are rolled, How many ways can this be done? How many ways are there to roll of sum of an 5 on the two dice? How many ways are there to roll a three on the first die? What is the probability of rolling a 3 on the first die or a sum of 5 on both dice? Give your answer as a completely simplified fraction using the " Γ " key

Answer 1

Explanation:

The probability that Ameer pulls out a yellow marble is 7/22 (since there are 7 yellow marbles out of a total of 22 marbles).

After Ameer pulls out a yellow marble, there are now 6 yellow marbles left out of a total of 21 marbles.

The probability that Becky pulls out a yellow marble (without replacement) is 6/21.

To find the probability that both Ameer and Becky pull out a yellow marble, we multiply the probabilities:

P(both yellow) = (7/22) * (6/21) = 42/462 ≈ 0.091

The probability of pulling out a blue marble is 18/39, and the probability of pulling out a green marble is 10/39.

Since the marbles are replaced after each draw, the probability of pulling out a blue and then a green marble (in any order) is:

P(blue and green) = P(blue) * P(green) + P(green) * P(blue)

= (18/39) * (10/39) + (10/39) * (18/39)

= 180/1521 + 180/1521

= 360/1521 ≈ 0.237

For the multiple-choice questions, each question can be answered in 3 ways (since there are 3 possible answers).

So, the number of ways to answer the multiple-choice questions is 3^7.

For the true-false questions, each question can be answered in 2 ways (since there are 2 possible answers).

So, the number of ways to answer the true-false questions is 2^4.

To find the total number of ways to answer all the questions, we multiply the possibilities:

Total number of ways = 3^7 * 2^4 = 2187 * 16 = 35,112

The number of ways to choose 3 members from a group of 30 is given by the combination formula:

Number of ways = C(30, 3) = 30! / (3! * (30-3)!) = 30! / (3! * 27!)

= (30 * 29 * 28) / (3 * 2 * 1) = 4060

Since all the pears are identical and all the apples are identical, the number of ways to line up the fruit can be found using the permutation formula:

Number of ways = P(10, 5) = 10! / (10 - 5)!

= 10! / 5!

= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)

= 3024

The expected value of playing one game of Moola is -$0.03. So, the expected gain from playing 7678 games can be found by multiplying the expected value by the number of games:

Expected gain = -$0.03 * 7678

= -$230.34

The probability that Juan does not make his next free throw is equal to 1 minus the probability that he makes it:

P(not making the free throw) = 1 - P(making the free throw)

= 1 - (71/85)

= (85/85) - (71/85)

= 14/85 ≈ 0.165

The number of ways to choose a President, Vice President, and Treasurer from a group of 19 members is given by the permutation formula:

Number of ways = P(19, 3) = 19! / (19 - 3)!

= 19! / 16!

= (19 * 18 * 17) / (3 * 2 * 1)

= 969

The number of ways to roll two fair six-sided dice is 6^2 = 36.

The number of ways to roll a sum of 5 on the two dice is 4 (since there are four combinations: (1, 4), (4, 1), (2, 3), (3, 2)).

The number of ways to roll a three on the first die is 1 (only one combination: (1, x)).

To find the probability of rolling a 3 on the first die or a sum of 5 on both dice, we add the probabilities:

P(rolling a 3 on the first die or a sum of 5 on both dice) = P(rolling a 3 on the first die) + P(rolling a sum of 5 on both dice)

= 1/6 + 4/36

= 6/36 + 4/36

= 10/36

= 5/18